L(s) = 1 | + (−1.40 + 0.147i)2-s + (−0.669 + 0.743i)3-s + (0.978 − 0.207i)4-s + (0.104 − 0.994i)5-s + (0.831 − 1.14i)6-s + (−1.05 + 0.946i)7-s + (−0.104 − 0.994i)9-s + 1.41i·10-s + (−0.500 + 0.866i)12-s + (1.33 − 1.48i)14-s + (0.669 + 0.743i)15-s + (−0.913 + 0.406i)16-s + (0.294 + 1.38i)18-s + (−0.104 − 0.994i)20-s − 1.41i·21-s + ⋯ |
L(s) = 1 | + (−1.40 + 0.147i)2-s + (−0.669 + 0.743i)3-s + (0.978 − 0.207i)4-s + (0.104 − 0.994i)5-s + (0.831 − 1.14i)6-s + (−1.05 + 0.946i)7-s + (−0.104 − 0.994i)9-s + 1.41i·10-s + (−0.500 + 0.866i)12-s + (1.33 − 1.48i)14-s + (0.669 + 0.743i)15-s + (−0.913 + 0.406i)16-s + (0.294 + 1.38i)18-s + (−0.104 − 0.994i)20-s − 1.41i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3347788400\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3347788400\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.669 - 0.743i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (1.40 - 0.147i)T + (0.978 - 0.207i)T^{2} \) |
| 5 | \( 1 + (-0.104 + 0.994i)T + (-0.978 - 0.207i)T^{2} \) |
| 7 | \( 1 + (1.05 - 0.946i)T + (0.104 - 0.994i)T^{2} \) |
| 13 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-1.05 + 0.946i)T + (0.104 - 0.994i)T^{2} \) |
| 31 | \( 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)T^{2} \) |
| 37 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2} \) |
| 61 | \( 1 + (-0.575 - 1.29i)T + (-0.669 + 0.743i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 83 | \( 1 + (-0.575 - 1.29i)T + (-0.669 + 0.743i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.921247544726170095275067037184, −9.388686352880850786650102192806, −8.792578243524450692268478327474, −8.165125204619127871632063814348, −6.79105259964669047978160623219, −6.13174075558074059568767009668, −5.15013664563475421932767427550, −4.22890745303441996902248745392, −2.75035842885963088004696546835, −0.980082450101078726749206518329,
0.71242096520896867364893159056, 2.17689025838903540195740698657, 3.33303969803557289895721908613, 4.83881084348116753401390842820, 6.34005870438598094279381739240, 6.73909424168774972427113010339, 7.42467667958372387939684680710, 8.155934506703413495188785799577, 9.278540426581106116447468862601, 10.09006533517694229200863166500